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We consider a low energy effective theory of $p$-branes in a $D$-dimensionalspacetime, and impose two conditions: 1) the theory is scale invariant, and 2)the electric-magnetic dual $(D-p-4)$-branes exist and they obey the same typeof interactions to the $p$-branes. (We also assume other natural conditionssuch as Lorentz invariance but not string theory, supersymmetry, supergravityand so on.) We then ask what $p$ and $D$ are consistent with these conditions.Using simple dimensional analysis, we find that only two solutions arepossible: $(p,D)=(2,11)$ and $(p,D)=(2n-1,4n+2)$, ($n=1,2,3,\cdots$). The firstsolution corresponds to M-theory, and the second solutions at $n=1$ and $n=2$correspond to self-dual strings in little string theory and D3-branes in typeIIB superstring theory, respectively, while the second solutions for $n \ge 3$are unknown but would be higher spin theories. Thus, quantum gravity (masslessspin two theory) satisfying our two conditions would only be superstringtheories, and the conditions would be strong enough to characterize superstringtheories in quantum gravity.

Constraining Spacetime Dimensions in Quantum Gravity by Scale Invariance and Electric-Magnetic Duality